part c asks us to find the "actual profit function" when the profit for producing 2500 items is 19,750 dollars (Note: 19750 becomes 19.75 when we plug into the function)...

Then part d asks us what is the max profit to be made and what level of production is needed to acheive it?

2. Relevant equations

3. The attempt at a solution

After substitution, the function for part c is as follows:
p(x)= (-29/9)x²+(172/9)x-(71/9) thousand dollars
such that 19.75= (-29/9)x²+(172/9)x-(71/9) thousand dollars
I'm guessing we have to use this formula to answer part d...I have solved x, where, in this case, x=0.44 or 5.48
Yet, I don't think these x-values will help me solve part d?!

Staff: Mentor

a=t=(-29/9)...hmm we have done some calculus...but we haven't applied it yet to functions in math. In physics, I have applied calculus when dealing with acceleration and velocity. I'm assuming it's the same idea.

Staff: Mentor

a=t=(-29/9)...hmm we have done some calculus...but we haven't applied it yet to functions in math. In physics, I have applied calculus when dealing with acceleration and velocity. I'm assuming it's the same idea.

The derivative of a function is the slope of the function at each point. So for example, if y = -2x^2, then dy/dx = -4x. So if you plot y(x) and dy/dx, you will see that the function y(x) is an upside-down parabola centered on the origin, and the slope function dy/dx is positive for -x and negative for +x, and it is zero at the origin where the function is maximum.

So the general technique for finding maxima and minima of a function is to take the deriviate of the function and set that equal to zero. Solving that equation gives you all the values of x where the function has a max or min. You then either have to plot the function to see if it's a max or min, or else take the second derivative to see if the funtion has positive curvature (like a cup shape) or negative curvature (like an umbrella shape) at each max/min. Make sense?

This part is rather confusing "So if you plot y(x) and dy/dx, you will see that the function y(x) is an upside-down parabola centered on the origin, and the slope function dy/dx is positive for -x and negative for +x, and it is zero at the origin where the function is maximum"
But I understood the rest!
Based on your comment, I can simply graph the quadratic equation (0= (-29/9)x²+(172/9)x-(995/36)) then find the coordinates of the peak which represent the max. profit. I plug in the values which will enable me to find the max. profit! Using differentiation, the equation will be: (-58/9)x+(172/9)...right?

The simplest way, especially if youre not familliar with calc, is to find the axis of symmetry and plug that in to get the y coordinate of the vertex. The calc comes in handy for higher degree polynomials